(ε, δ)-definition of limit: Difference between revisions

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In [[fluid dynamics]], the '''Hadamard–Rybczynski equation''' gives the [[terminal velocity]] of slowly moving [[spherical]] [[liquid bubble|bubble]] through an ambient [[fluid]]. It is named after [[Jacques Hadamard]] and [[Witold Erazm Rybczynski|Witold Rybczynski]]:


:<math> W_b = \frac{2}{3} \frac {R^2 g (\rho_b - \rho_o)}{\mu_o}
\frac {\mu_o + \mu_b}{2\mu_o + 3\mu_b}
</math>
where
* <math>R</math> is the radius of the bubble.
* <math>g</math> the gravitational acceleration.
* <math>\rho_b</math> the density of the bubble.
* <math>\rho_o</math> the density of the ambient fluid.
* <math>\mu_b</math> the viscosity of the bubble.
* <math>\mu_o</math> the viscosity of the ambient fluid.


The Hadamard–Rybczynski equation can be derived from the [[Navier–Stokes equations]] by considering only the [[buoyancy force]] and [[drag force]] acting on the moving bubble. The surface tension force and inertia force of the bubble are neglected.<ref>{{cite book |last=Clift, R. C., Grace, B. J., and Weber, M. E.,|first= |title=Bubbles, Drops, and Particles |publisher=[[Dover Publications]] |year=2005 |isbn=978-0-486-44580-9}}
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</ref>
 
==References==
{{reflist}}
 
==Further reading==
*{{Cite journal |first=J. S. |last=Hadamard |title={{lang|fr|Mouvement permanent lent d'une sphere liquide et visqueuse dans un liquide visqueux}} |journal=CR Acad. Sci. |volume=152 |year=1911 |pages=1735 }}
*{{Cite journal |first=W. |last=Rybczynski |title={{lang|de|Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium}} |journal=Bull. Acad. Sci. Cracovi, A. |year=1911 |pages=40–46 }}
 
{{DEFAULTSORT:Hadamard-Rybczynski equation}}
[[Category:Fluid dynamics]]
[[Category:Bubbles]]
[[Category:Equations of fluid dynamics]]
[[Category:Buoyancy]]

Latest revision as of 19:27, 11 January 2015


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